This paper reports the technique for determining the thickness profile (i.e., the variation of thickness as a function of radial distance) for a deformable glass mirror whose inner surface must always maintain a spherical shape. The convex mirror produced by the periodic deformation (i.e., from optically flat to maximally spherical deformation of the optical surface at a given sinusoidal frequency) has associated with it a dynamically varying focal spot. The mirror is designed to assume a convex chape which is perfectly spherical over its deformation range, thus providing a varying negative focal point which can be incorporated into an associated optical system making use of the fluctuating focal position. The mirror thickness profile is determined by solving the equation of motion for a variable thickness plate with the constraint that the deformed surface be perfectly spherical when a uniform load is applied to the back surface of the plate. The final result gives the thickness as a function of radius with load, frequency, and physical constants of the plate as parameters. This work has significance in the field of adaptive optics.
Mark A. Massie,
Arthur W. Leissa,
"Mirror Thickness Profile Determination For A Variable Focus Device," Optical Engineering 21(2), 212301 (1 April 1982). https://doi.org/10.1117/12.7972899